Subgroups — C₇₃×#6860 #31801154

Order 17739.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₇₃	C₃	1	2	yes	maximal
2	C₇₃×C₃²⋊C₉	C₃	1	2	yes	maximal
3	C₃³×C₇₃	C₃²	1	1	yes	characteristic
4	3₋¹⁺²×C₇₃	—	3	6	no	—
5	C₃×C₉×C₇₃	—	3	2	no	—
6	C₃×C₉×C₇₃	—	3	2	no	—
7	C₃²×C₇₃	—	3	2	no	—
8	C₃²×C₇₃	—	3	2	no	—
9	C₃²×C₇₃	H(3,𝔽₃)	1	1	yes	characteristic
10	C₉×C₇₃	—	3	6	no	—
11	C₉×C₇₃	—	9	2	no	—
12	#6860	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×C₇₃	—	9	1	no	—
14	C₃×C₇₃	AΣL(1,𝔽₂₇)	1	2	yes	—
15	C₃×C₇₃	#733	1	2	yes	—
16	C₃×3₋¹⁺²	C₃×C₇₃	1	2	yes	—
17	C₃²⋊C₉	C₃×C₇₃	1	2	yes	—
18	C₇₃	#6860	1	1	yes	characteristic, direct product factor, Sylow
19	C₃³	C₃²×C₇₃	1	1	yes	characteristic
20	3₋¹⁺²	—	3	6	no	—
21	C₃×C₉	—	3	2	no	—
22	C₃×C₉	—	3	2	no	—
23	C₃²	—	3	2	no	—
24	C₃²	—	3	2	no	—
25	C₃²	H(3,𝔽₃)×C₇₃	1	1	yes	characteristic
26	C₉	—	3	6	no	—
27	C₉	—	9	2	no	—
28	C₃	—	9	1	no	—
29	C₃	C₇₃×AΣL(1,𝔽₂₇)	1	2	yes	—
30	C₃	C₇₃×#733	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₇₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃²×C₇₃
prime radical	C₃³×C₇₃	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₇₃ ◁ C₃³×C₇₃ ◁ G

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. #6860	C₇₃	direct (N × H)
18. C₇₃	#6860	direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.